Convolution Process Revisited in Finite Location Mixtures and GARFISMA Long Memory Time Series

This article focuses on the utilisation of the convolution process by specialist theoretical statisticians representing two distinct analytical domains of the subject classified as nonparametric statistics and time series analysis. Main contribution is established through a discussion centered around some comparative properties of the convolution process, where the relevant operator performs similar yet different operations on differing statistical operands. As specific cases, examples from the considered subject areas in theoretical statistics known as finite location mixture distributions and fractionally integrated Gegenbauer autoregressive moving average (GARMA) seasonal long memory time series are presented to illustrate the overall value of convolution in arriving at closed form solutions. Importance and robustness of the process are revelations highlighted in the article as a secondary contribution.